Relevant background documents include:    1. K. Halbach, “Design of permanent multipole magnets with oriented rare earth cobalt material,” Nuclear Instruments and Methods 169, 1, 1980.    2. J. Mallinson, “One-sided fluxes—a magnetic curiosity?” IEEE Transactions on Magnetics 9, 678, 1973.    3. F. Bertora, A. Trequattrini, M. G. Abele, and H. Rusinek, “Shimming of yokeless permanent magnets designed to generate uniform fields,” Journal of Applied Physics 73, 6864, 1993.    4. E. Danieli, J. Mauler, J. Perlo, B. Blumich, and F. Casanova, “Mobile sensor for high resolution NMR spectroscopy and imaging, Journal of Magnetic Resonance 198, 80, 2009.    5. E. Lord, “Tiling space with regular and semi-regular polyhedra,” http://met.iisc.ernet.in/˜lord/webfiles/clusters/andreini.pdf, accessed May 29, 2013.    6. F. Bloch, O. Cugat, G. Meunier, J. Toussaint, “Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 tesla permanent magnet flux source,” IEEE Transactions on Magnetics 34, 2465, 1998.    7. U.S. Pat. No. 4,837,542 to H. Leupold, “Hollow substantially hemispherical permanent magnet high-field flux source for producing a uniform high field,” 1989.    8. U.S. Patent Application 2011/0137589, G. Leskowitz, G. McFeetors, and S. Pernecker, “Method and apparatus for producing homogeneous magnetic fields,” 2011.    9. U.S. Pat. No. 7,373,716 to Ras, “Method for Constructing Permanent Magnet Assemblies,” 2008.    10. N. A. Leupold et al., Journal of Applied Physics, vol 87, no 9, p. 4730-4 (2000)    11. J. Chen, Y Zhang, and J Xiao, “Design and analysis of the novel test tube magnet for portable NMR device,” Progress in Electromagnetics Research Symposium (PIERS) Online, 3 (6), 900-904 (2007).
One design for producing a substantially strong magnetic field in a small volume is the Halbach cylinder, wherein magnetic dipoles within high-coercivity permanent magnet materials are arranged around a central cavity. FIG. 1 shows a cross-sectional view of an idealization of a Halbach cylinder 10, along with a coordinate system that is used to compute and select the orientations of magnetic dipoles, shown as arrows 11, within a region surrounding a central volume 12. In the idealized Halbach cylinder, magnetization direction {circumflex over (m)} is position-dependent according to the equation{circumflex over (m)}(ρ,θ,z)=cos(kθ){circumflex over (ρ)}+sin(kθ){circumflex over (θ)}in cylindrical polar coordinates ρ, θ, z, with integer parameter k=1 for the most prevalent case, which produces a substantially uniform field in the central volume 12. Other choices of k provide different, non-uniform field configurations. In practical implementations, discrete component magnets are used, as an approximation to the continuously varying magnetization suggested by FIG. 1.
FIGS. 2A, 2B, and 2C show example prior-art implementations of Halbach-cylinder-based magnet configurations. FIG. 2A, adapted from Bertora et al., shows a cylindrical configuration of magnets designated 20 surrounding space 24, that makes efficient use of space but employs many oblique shapes 21, 22, 23 in its design. FIG. 2B, adapted from Danieli, is an array 30 that uses simple shapes 31 to enclose space 32 but suffers from low packing density. When the space surrounding a central volume is broken up into regions, the individual component magnets placed therein may exhibit oblique shapes, such as those shown in FIG. 2A, that are difficult or expensive to fabricate with high tolerance. The magnetizations required within the component magnets may also be difficult to control with precision sufficient to ensure the quality of the magnetic field within the central volume. If, instead, simpler component magnets such as cubes are used, as in FIG. 2B, these can be fabricated and magnetized with high precision straightforwardly, but the geometrical constraints for some designs can result in a low packing density, with an attendant reduction in the field strength that can be produced. FIG. 2C is a cross section of an embodiment of a Halbach cylinder 40 comprising an array of closely packed hexagonal prisms 41 surrounding central space 42, disclosed in Leskowitz et al., U.S. Patent Application 2011/0137589.
In a Halbach-cylinder model the design ideal is an infinitely long cylinder. In practice, the cylinder is of finite length, which can lead to various technical problems and undesirable features in the primary magnetic field of the array, and designs attempting to overcome these disadvantages can be complex. An alternative approach for producing homogeneous fields therefore uses a Halbach sphere, practical embodiments of which have been suggested by Leupold.
FIG. 3A shows a sphere 50 enclosing a central cavity 51 and having local magnetic dipole orientations 52. Once a desired magnetic field axis, {circumflex over (v)}, is selected, the required magnetization directions for the component magnets in the assembly can be calculated by establishing a spherical polar coordinate system with colatitude angle θ=0 along the magnetic field direction, then calculating the magnetization direction for the given magnet's center coordinates.
In order to best approximate a uniform field in the idealized case, magnetization direction {circumflex over (m)} within the spherical shell surrounding the central cavity is position-dependent according to the equation{circumflex over (m)}(r,θ,ϕ)=cos(kθ){circumflex over (r)}+sin(kθ){circumflex over (θ)}in spherical polar coordinates r, θ, ϕ, again with parameter k=1 for the uniform-field case. It will be observed that magnetization in the spherical case differs from the magnetization in the cylindrical case. In the Halbach sphere model, the magnetization of the dipole at a position {right arrow over (r)}=r{circumflex over (r)} lies in the meridional plane spanned by {circumflex over (r)} and {circumflex over (θ)}, but in the Halbach cylinder model, the magnetization lies in a plane spanned by {circumflex over (ρ)}=r{circumflex over (r)}−z{circumflex over (z)} and {circumflex over (θ)}, the former unit vector being the one directed away from the cylindrical symmetry axis. In particular, in the idealized Halbach cylinder case, the magnetization direction has no {circumflex over (z)} component (along the cylindrical symmetry axis) and is independent of the z coordinate of the dipole's position. A variety of numerical representations of such position-dependent magnetizations are possible and will be readily identified and understood by those skilled in the art.
Such spherical assemblies are generally composed of combinations of magnets having complex shapes, as illustrated in FIG. 3B, adapted from Leupold. In FIG. 3B. it will be seen that the sphere 60 comprises multiple component primary magnets 61 having chosen dipole orientations 62 and surrounding central cavity 63. In order to achieve the desired conformation and field, a large number of different primary magnets having different shapes and magnetic orientations is required.